A response spectrum method for stationary random vibration analysis of linear structures is developed. The method is based on the assumption that the input excitation is a wide-band stationary Gaussian process and the response is also stationary. However, it can also be used as a good approximation for the response to a transient stationary Gaussian input with a duration several times longer than the fundamental period of the structure. Various response quantities, including the mean-squares of the response and its time derivative, the response mean frequency, and the cumulative distribution and the mean and variance of the peak response are obtained in terms of the ordinates of the mean response spectrum of the input excitation and the modal properties of the structure. The formulation includes the cross-correlation between modal responses, which are shown to be significant for modes with closely spaced natural frequencies. The proposed procedure is demonstrated for an example structure that is subjected to earthquake induced base excitations. Computed results based on the response spectrum method are in close agreement with simulation results obtained from time-history dynamic analysis. The significance of closely spaced modes and the error associated with a conventional method neglects the modal correlations are also demonstrated through this example.